Iterative Algorithms for Single-Valued and Multi-Valued No expansive-Type Mappings in Real Lebesgue Spaces.


Algorithms for single-valued and multi-valued nonexpansive-type mappings have continued to attract a lot of attentions because of their remarkable utility and wide applicability in modern mathematics and other reasearch areas,(most notably medical image reconstruction, game theory and market economy).

The first part of this thesis presents contributions to some crucial new concepts and techniques for a systematic discussion of questions on algorithms for single- valued and multi-valued mappings in real Hilbert spaces.

Novel contributions are made on iterative algorithms for fixed points and solutions of the split equality fixed point problems of some single-valued pseudocontractive-type mappings in real Hilbert spaces.

Interesting contributions are also made on it- erative algorithms for fixed points of a general class of multivalued strictly pseu- docontractive mappings in real Hilbert spaces using a new and novel approach and the thorems were gradually extended to a countable family of multi-valued mappings in real Hilbert spaces.

It also contains contains original research and important results on iterative approximations of fixed points of multi-valued tempered Lipschitz pseudocontractive mappings in Hilbert spaces.

Apart from using some well known iteration methods and identities, some very new and innovative iteration schemes and identities are constructed.

The thesis serves as a basis for unifying existing ideas in this area while also gener- alizing many existing concepts. In order to demonstrate the wide applicability of the theorems.


Background of Study

Fixed Point Theory is concerned with solutions of the equation x = Tx                              

where T is a (possibly) nonlinear operator defined on a metric space. Any x that solves (1.0.1) is called a fixed point of T and the collection of all such elements is denoted by F (T ). For  a multi-valued  mapping : 2X, a  fixed point of T is any x in X such that x Tx.

Fixed Point Theory is inarguably the most powerful and effective tools used in modern nonlinear analysis today. It is still an area of current intensive research as it has vast applicability in establishing existence and uniqueness of solutions of diverse mathematical models like solutions to optimization prob- lems, variational analysis, and ordinary differential equations.

These models represent various phenomena arising in different fields, such as steady state temperature distribution, neutron transport theory, economic theories, chem- ical equations, optimal control of systems, models for population, epidemics and flow of fluids.

Then finding a solution to the initial value problem (1.0.2) amounts to finding a fixed point of T .The existence(and uniqueness) of solution to equation (1.0.1), certainly, de- pends on the geometry of the space and the nature of the mapping T .

Ex- istence theorems are concerned with establishing sufficient conditions under which the equation (1.0.1) will have a solution, but does not neccesarily show how to find them.


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Alber, “Metric and generalized projection operators in Banach spaces: properties and applications”. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type (A. G. Kartsatos, Ed.), Marcel Dekker, New York (1996), pp. 15-50.

I. Alber and S. Reich, “An iterative method for solving a class of nonlinear operator equations in Banach spaces”, Panamer. Math. J., 4(2) (1994), 39-54.

Alber and S, Guerre-Delabriere, “On the projection methods for fixed point problems”, Analysis (Munich), vol. 21 (2001), no. 1, pp. 17-39.

Alber and I. Ryazantseva,“ Nonlinear Ill Posed Problems of Monotone Type”, Springer, London, UK, (2006).

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